3.777 \(\int \frac{\sqrt{a+c x^4}}{x^8} \, dx\)

Optimal. Leaf size=129 \[ -\frac{c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+c x^4}}-\frac{\sqrt{a+c x^4}}{7 x^7}-\frac{2 c \sqrt{a+c x^4}}{21 a x^3} \]

[Out]

-Sqrt[a + c*x^4]/(7*x^7) - (2*c*Sqrt[a + c*x^4])/(21*a*x^3) - (c^(7/4)*(Sqrt[a]
+ Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c
^(1/4)*x)/a^(1/4)], 1/2])/(21*a^(5/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.100734, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+c x^4}}-\frac{\sqrt{a+c x^4}}{7 x^7}-\frac{2 c \sqrt{a+c x^4}}{21 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x^8,x]

[Out]

-Sqrt[a + c*x^4]/(7*x^7) - (2*c*Sqrt[a + c*x^4])/(21*a*x^3) - (c^(7/4)*(Sqrt[a]
+ Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c
^(1/4)*x)/a^(1/4)], 1/2])/(21*a^(5/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 10.8297, size = 116, normalized size = 0.9 \[ - \frac{\sqrt{a + c x^{4}}}{7 x^{7}} - \frac{2 c \sqrt{a + c x^{4}}}{21 a x^{3}} - \frac{c^{\frac{7}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{21 a^{\frac{5}{4}} \sqrt{a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(1/2)/x**8,x)

[Out]

-sqrt(a + c*x**4)/(7*x**7) - 2*c*sqrt(a + c*x**4)/(21*a*x**3) - c**(7/4)*sqrt((a
+ c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*elliptic_f(2*at
an(c**(1/4)*x/a**(1/4)), 1/2)/(21*a**(5/4)*sqrt(a + c*x**4))

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Mathematica [C]  time = 0.347415, size = 106, normalized size = 0.82 \[ \frac{-\frac{3 a^2}{x^7}+\frac{2 i c^2 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}-\frac{5 a c}{x^3}-2 c^2 x}{21 a \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^8,x]

[Out]

((-3*a^2)/x^7 - (5*a*c)/x^3 - 2*c^2*x + ((2*I)*c^2*Sqrt[1 + (c*x^4)/a]*EllipticF
[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(21*a*S
qrt[a + c*x^4])

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Maple [C]  time = 0.02, size = 110, normalized size = 0.9 \[ -{\frac{1}{7\,{x}^{7}}\sqrt{c{x}^{4}+a}}-{\frac{2\,c}{21\,a{x}^{3}}\sqrt{c{x}^{4}+a}}-{\frac{2\,{c}^{2}}{21\,a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x^8,x)

[Out]

-1/7*(c*x^4+a)^(1/2)/x^7-2/21*c*(c*x^4+a)^(1/2)/a/x^3-2/21*c^2/a/(I/a^(1/2)*c^(1
/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^
4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + a}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^8,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + a)/x^8, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + a}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^8,x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + a)/x^8, x)

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Sympy [A]  time = 3.70851, size = 46, normalized size = 0.36 \[ \frac{\sqrt{a} \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(1/2)/x**8,x)

[Out]

sqrt(a)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), c*x**4*exp_polar(I*pi)/a)/(4*x*
*7*gamma(-3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + a}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^8,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + a)/x^8, x)